Our comments regarding monitoring a trial are made within the context of boundaries constructed using the Lan-Demets
procedure, [1]}. Construction of the efficacy boundary is done under the null hypothesis that the drift
function is identically zero and can be done without appealing to the results presented here. If a futility boundary is
specified in the design then under either of the shape assumptions, one can apply the corresponding corollary
REF  or corollary REF  to calculate the drift function at each interim analysis which is
required to compute the futility boundary under the Lan-Demets approach [1]}. Note that the shape assumption
being made must be part of the interim analysis plan design. In the following discussion we will assume that the optimal
weighting shape condition REF  was specified in the design so that the discussion focuses on the application
of corollary REF . In this case, \(\beta ^{\star }\)  is the weighted average logged relative risk for which
the study is powered to detect and must also be specified in the interim analysis plan design. The values of
\(v(\tau )=\langle Q | {I}\hspace{-3.00003pt}{F}| Q \rangle _{\tau }\)  and \(m(\tau )=\langle Q | {I}\hspace{-3.00003pt}{F}| 1 \rangle _{\tau }\)  at the planned termination
of the study, \(\tau \) , must also be specified in the interim analysis plan design. We demonstrate in appendix
REF  when the only source of censoring is administrative censoring or other cause mortality,
how these functionals can be projected for a specific choice of weighting function, \(Q\) , based upon projected values of
the cross-arm pooled cumulative hazard function at several landmark times on study. We remark here that following
consensus, we recommend using a non-binding futility boundary which is constructed after construction of an efficacy
boundary which ignores the existence of the futility boundary. This is preferred to the joint construction of efficacy
and futility boundaries as that approach results in a discounted efficacy criterion.
